Download A = {a: for some b (a,b) О R}

Matthew Burke and Katherine Walton
Foundations of the Real
Number System
Defn: A model of Peano’s Axioms is a set N together with a function f and an object e (N,f,e) such
that,
(P1)
(P2)
(P3)
(P4)
(P5)
e N.
The domain of f is in N and for each x N, f(x) N.
If x N, then f(x) e.
If x, y N and f(x) = f(y), then x = y.
If A is a subset of N which contains e and contains f(x) for every x in A then A = N.
Any two models of Peano’s axioms are isomorphic.
Using the standard axioms of ZF set theory, we can construct a model which satisfies the Peano
axioms.
Without going into the detains, we will sketch the argument, and give relevant definitions.
We will consider only sets as elements of sets.
Defn: An ordered pair (a,b) is defined as the set {{a}, {a,b}}.
Note (a,a) = {{a}}.
It is true in ZF that if (a,b) = (x,y) then a = x, and b = y.
Defn: Cartesian Product: Given two sets A and B
A x B = {y: y = (a,b) for some a A, and for some b B}.
Defn: A set R is a binary relation iff every element of R is an ordered pair.
Note that every binary relation is a subset of a Cartesian product A x B where
A = {a: for some b (a,b) R}
B = {b: for some a (a,b) R}.
A is called the domain of R and B is the image of R.
Defn: An equivalence relation is a relation that is reflexive, symmetric, and transitive.
Defn: given a set x, denote by x+ the set x {x}. x+ is called the successor of x.
In ZF no set contains itself, so x+ is always distinct from x.
In ZF the null set, exists by axiom.
We can now define the first few natural numbers as follows:
0 
1  = {} = {}
2 {} = {

Before we define all of the natural numbers, we need a few more definitions.
Defn: A set x is a successor set if x, and if for each y x, y+  x.
The axiom of infinity asserts the existence of just such a set.
Theorem: There is a minimal successor set, i.e. a successor set that is a subset of every other
successor set.
Denote this minimal successor set by 

Define the natural numbers to be the elements of 
It is necessary to verify that the Peano Axioms are satisfied by (+,
Define addition for the natural numbers as follows:
m,n 
m + 0 m, and
m + n+  (m + n)+
Multiplication (note we will often write mn instead of m x n):
m,n 
m x 0 0, and
m x n+ m + (m x n).
Things to show for ‘+’ (note almost all proofs about N are done by induction):
(i) n  
0+n=n
(ii) m,n 
m+ + n = (m + n)+
(iii)m,n 
m+n=n+m
(iv) m,n,p 
m + (n + p) = (m + n) + p
Things to show for ‘x’:
(i) n  
(ii) m,n 
(iii)m,n 
0xn=0
m+ x n = (m x n) + n
mxn=nxm
(iv) m,n,p 
(v) m,n,p 
m x (n + p) = (m x n) + (m x p)
m x (n x p) = (m x n) x p
Note special properties of 0 and 1:
0+m=m=m+0
0xm=0=mx0
m 
m 

Recall 0+ = 1 so m x 1 = m x 0+ = m + (m x 0) = m m 

Similarly, 1 x m = m m 

So 0 behaves like an additive identity, and 1 behaves like a multiplicative identity.
Note we will now begin using N instead of  to denote the natural numbers.
Defn: m, n N, m  n iff x N with x  0 s.t.
m + x = n.
Defn: m, n N, m n iff x N s.t.
m + x = n.
Properties of N:
(i)
n N either n = 0, of m N s.t. n = m+.
(ii)
m, n N if m  0, then m + n  0.
(iii)
m, n N if m  0 and n  0, then m x n  0.
(iv)
n N, n+ = n + 1.
(v)
m, n N, if m  n then m+ n.
(vi)
m, n, p N, if m + n = p + n, then m = p (cancellation of addition).
Defn: A set A of natural numbers has a least member iff n A s.t. n m m, n A. If A
has a least member, then it is n.
Well Ordering Principle: Every non-empty set of Natural numbers has a least member.
Integers
Let a, b, c, d N. Then define an equivalence relation on N by
(a,b) ~ (c,d) iff a + d = b + c
Intuitively we have
a–b=c–d
iff
a + d = b + c.
Then we define the integers to be the equivalence classes under ~.
Ex. The set {(a,b): a + 1 = b} is the integer –1. Note (2,3)  –1, and 2 – 3 = –1.
We will use (a,b) to denote the equivalence class determined by (a,b).
Define ‘’ in the integers by
(a,b) (c,d)  (a + b,c +d)
Define ‘’ in the integers by
(a,b) (c,d)  (ac + bd,ad +bc)
Verify and are well defined.
To Show:
Commutative, and Associative properties of 


Commutative, Associative, and Distributive properties of 

Note the special properties of (0,0), and (1,0).
(0,0) is the additive identity in the integers.
(1,0) is the multiplicative identity in the integers.
Denote the integers by Z.
N is embedded in Z. (m,0) Z behaves like m N.
Additive Inverses:
Given an arbitrary integer (a,b) note that (a,b)  (b,a) = (a + b,b + a) = (0,0)
We write – (a,b) for (b,a), and we abbreviate (a,b) (c,d)) by (a,b) (c,d).
To Show:
(i)
(–(a,b)) (c,d) (a,b) (c,d)).
(ii)
(–(a,b)) (c,d)) = (a,b) (c,d).
Defn: An integer (a,b) is positive iff b < a in N.
We need to verify that positive integers are well defined.
Denote the set of positive integers by Z+.
More properties of Z:
(i)
 x  Z, either x  Z+, x = 0, or x  Z+.
(ii)
 x,y  Z+ x  y  Z+.
(iii)
 x,y  Z+ x  y  Z+.
(iv)
 x,y  Z, if x  0 and y  0, then x  y  0.
Thus Z is an integral domain.